Theorem opid 3880

Description: The ordered pair ⟨𝐴, 𝐴⟩ in Kuratowski's representation. (Contributed by FL, 28-Dec-2011.)

Hypothesis

Ref	Expression
opid.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
opid	⊢ ⟨𝐴, 𝐴⟩ = {{𝐴}}

Proof of Theorem opid

Step	Hyp	Ref	Expression
1		dfsn2 3683	. . . 4 ⊢ {𝐴} = {𝐴, 𝐴}
2	1	eqcomi 2235	. . 3 ⊢ {𝐴, 𝐴} = {𝐴}
3	2	preq2i 3752	. 2 ⊢ {{𝐴}, {𝐴, 𝐴}} = {{𝐴}, {𝐴}}
4		opid.1	. . 3 ⊢ 𝐴 ∈ V
5	4, 4	dfop 3861	. 2 ⊢ ⟨𝐴, 𝐴⟩ = {{𝐴}, {𝐴, 𝐴}}
6		dfsn2 3683	. 2 ⊢ {{𝐴}} = {{𝐴}, {𝐴}}
7	3, 5, 6	3eqtr4i 2262	1 ⊢ ⟨𝐴, 𝐴⟩ = {{𝐴}}

Syntax hints:   = wceq 1397   ∈ wcel 2202  Vcvv 2802  {csn 3669  {cpr 3670  ⟨cop 3672

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676  df-op 3678

This theorem is referenced by:  dmsnsnsng  5214  funopg  5360  funopsn  5829